1.

Find Integrating factor of dy = {[e^(x-y)] ( e^x-e^y)} dx is

Answer»

Answer:

CORRECT option is

C

e

y

=c.exp(−e

X

)+e

x

−1

dx

dy

=e

x−y

(e

x

−e

y

)

dx

dy

=

e

y

e

x

(e

x

−e

y

)

e

y

dx

dy

=e

2x

−e

x

e

y

e

y

dx

dy

+e

x

e

y

=e

2x

Put e

y

=v

e

y

dx

dy

=

dx

dv

e

y

dx

dv

+ve

x

=e

2x

which is a linear differential eqn with v as dependent variable.

Here, P=e

x

;Q=e

2x

Integrating factor $$I.F.=e^\int{e^x}dx =e^{e^x}$$

So, the solution of GIVEN differential eqn is

v.e

e

x

=∫e

e

x

e

2x

dx+C

Put e

x

=t in the above integral

⇒e

x

dx=dt

v.e

e

x

=∫e

t

tdt+C

⇒e

y

e

e

x

=te

t

−e

t

+C

⇒e

y

e

e

x

=e

x

e

e

x

−e

e

x

+C

⇒e

y

=e

x

−1+Ce

−e

x


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